A334028 -id:A334028 - cp's OEIS frontend

A351015 Smallest k such that the k-th composition in standard order has n distinct runs.

0, 1, 5, 27, 155, 1655, 18039, 281975

Offset: 0

Gus Wiseman, Feb 10 2022

The n-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of n, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

It would be very interesting to have a formula or general construction for a(n). - Gus Wiseman, Feb 12 2022

			The terms together with their binary expansions and corresponding compositions begin:
       0:                    0  ()
       1:                    1  (1)
       5:                  101  (2,1)
      27:                11011  (1,2,1,1)
     155:             10011011  (3,1,2,1,1)
    1655:          11001110111  (1,3,1,1,2,1,1,1)
   18039:      100011001110111  (4,1,3,1,1,2,1,1,1)
  281975:  1000100110101110111  (4,3,1,2,2,1,1,2,1,1,1)

Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

The version for Heinz numbers and prime multiplicities is A006939.
Counting not necessarily distinct runs gives A113835 (up to zero).
Using binary expansions instead of standard compositions gives A350952.
These are the positions of first appearances in A351014.
A005811 counts runs in binary expansion, distinct A297770.
A011782 counts integer compositions.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A085207 represents concatenation of standard compositions, reverse A085208.
A333489 ranks anti-runs, complement A348612.
Counting words with all distinct runs:
- A351013 = compositions, for run-lengths A329739, ranked by A351290.
- A351016 = binary words, for run-lengths A351017.
- A351018 = binary expansions, for run-lengths A032020, ranked by A175413.
Selected statistics of standard compositions (A066099, reverse A228351):
- Length is A000120.
- Sum is A070939.
- Runs are counted by A124767.
- Number of distinct parts is A334028.
Cf. A106356, A238279, A242882, A318928, A325545, A328592, A329745, A351201, A351204.

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
s=Table[Length[Union[Split[stc[n]]]],{n,0,1000}];
Table[Position[s,k][[1,1]]-1,{k,Union[s]}]

A351016 Number of binary words of length n with all distinct runs.

Original entry on oeis.org

1, 2, 4, 6, 12, 18, 36, 54, 92, 154, 244, 382, 652, 994, 1572, 2414, 3884, 5810, 8996, 13406, 21148, 31194, 47508, 70086, 104844, 156738, 231044, 338998, 496300, 721042, 1064932, 1536550, 2232252, 3213338, 4628852, 6603758, 9554156, 13545314, 19354276

Offset: 0

Gus Wiseman, Feb 07 2022

nonn

These are binary words where the runs of zeros have all distinct lengths and the runs of ones also have all distinct lengths. For n > 0 this is twice the number of terms of A175413 that have n digits in binary.

			The a(0) = 1 through a(4) = 12 binary words:
  ()   0    00    000    0000
       1    01    001    0001
            10    011    0010
            11    100    0011
                  110    0100
                  111    0111
                         1000
                         1011
                         1100
                         1101
                         1110
                         1111
For example, the word (1,1,0,1) has three runs (1,1), (0), (1), which are all distinct, so is counted under a(4).

Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

The version for compositions is A351013, lengths A329739, ranked by A351290.
The version for [run-]lengths is A351017.
The version for expansions is A351018, lengths A032020, ranked by A175413.
The version for patterns is A351200, lengths A351292.
The version for permutations of prime factors is A351202.
A000120 counts binary weight.
A001037 counts binary Lyndon words, necklaces A000031, aperiodic A027375.
A005811 counts runs in binary expansion.
A011782 counts integer compositions.
A242882 counts compositions with distinct multiplicities.
A297770 counts distinct runs in binary expansion.
A325545 counts compositions with distinct differences.
A329767 counts binary words by runs-resistance.
A351014 counts distinct runs in standard compositions.
A351204 counts partitions whose permutations all have all distinct runs.
Cf. A003242, A098859, A106356, A116608, A238130 or A238279, A328592, A329738, A329745, A334028, A351201.

Table[Length[Select[Tuples[{0,1},n],UnsameQ@@Split[#]&]],{n,0,10}]

from itertools import groupby, product
def adr(s):
    runs = [(k, len(list(g))) for k, g in groupby(s)]
    return len(runs) == len(set(runs))
def a(n):
    if n == 0: return 1
    return 2*sum(adr("1"+"".join(w)) for w in product("01", repeat=n-1))
print([a(n) for n in range(20)]) # Michael S. Branicky, Feb 08 2022

a(n>0) = 2 * A351018(n).

a(25)-a(32) from Michael S. Branicky, Feb 08 2022

a(33)-a(38) from David A. Corneth, Feb 08 2022

A351596 Numbers k such that the k-th composition in standard order has all distinct run-lengths.

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 8, 10, 11, 14, 15, 16, 19, 21, 23, 26, 28, 30, 31, 32, 35, 36, 39, 42, 47, 56, 60, 62, 63, 64, 67, 71, 73, 74, 79, 84, 85, 87, 95, 100, 106, 112, 119, 120, 122, 123, 124, 126, 127, 128, 131, 135, 136, 138, 143, 146, 159, 164, 168, 170, 171

Offset: 1

Gus Wiseman, Feb 24 2022

nonn

The n-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of n, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The terms together with their binary expansions and corresponding compositions begin:
   0:      0  ()
   1:      1  (1)
   2:     10  (2)
   3:     11  (1,1)
   4:    100  (3)
   7:    111  (1,1,1)
   8:   1000  (4)
  10:   1010  (2,2)
  11:   1011  (2,1,1)
  14:   1110  (1,1,2)
  15:   1111  (1,1,1,1)
  16:  10000  (5)
  19:  10011  (3,1,1)
  21:  10101  (2,2,1)
  23:  10111  (2,1,1,1)

The version using binary expansions is A044813.
The version for Heinz numbers and prime multiplicities is A130091.
These compositions are counted by A329739, normal A329740.
The version for runs instead of run-lengths is A351290, counted by A351013.
A005811 counts runs in binary expansion, distinct A297770.
A011782 counts integer compositions.
A085207 represents concatenation of standard compositions, reverse A085208.
A333489 ranks anti-runs, complement A348612.
A345167 ranks alternating compositions, counted by A025047.
A351204 counts partitions where every permutation has all distinct runs.
Counting words with all distinct run-lengths:
- A032020 = binary expansions, for runs A351018.
- A351017 = binary words, for runs A351016.
- A351292 = patterns, for runs A351200.
Selected statistics of standard compositions (A066099, A228351):
- Length is A000120.
- Sum is A070939.
- Runs are counted by A124767, distinct A351014.
- Heinz number is A333219.
- Number of distinct parts is A334028.
Cf. A098859, A106356, A175413, A238279, A242882, A328592, A329745, A333628, A350952, A351015, A351202.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],UnsameQ@@Length/@Split[stc[#]]&]

A351290 Numbers k such that the k-th composition in standard order has all distinct runs.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 47, 48, 50, 51, 52, 55, 56, 57, 58, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 78

Offset: 1

Gus Wiseman, Feb 10 2022

nonn

The n-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of n, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The terms together with their binary expansions and corresponding compositions begin:
   0:      0  ()
   1:      1  (1)
   2:     10  (2)
   3:     11  (1,1)
   4:    100  (3)
   5:    101  (2,1)
   6:    110  (1,2)
   7:    111  (1,1,1)
   8:   1000  (4)
   9:   1001  (3,1)
  10:   1010  (2,2)
  11:   1011  (2,1,1)
  12:   1100  (1,3)
  14:   1110  (1,1,2)
  15:   1111  (1,1,1,1)

Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

The version for Heinz numbers and prime multiplicities is A130091.
The version using binary expansions is A175413, complement A351205.
The version for run-lengths instead of runs is A329739.
These compositions are counted by A351013.
The complement is A351291.
A005811 counts runs in binary expansion, distinct A297770.
A011782 counts integer compositions.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A085207 represents concatenation of standard compositions, reverse A085208.
A333489 ranks anti-runs, complement A348612.
A345167 ranks alternating compositions, counted by A025047.
A351204 counts partitions where every permutation has all distinct runs.
Counting words with all distinct runs:
- A351016 = binary words, for run-lengths A351017.
- A351018 = binary expansions, for run-lengths A032020.
- A351200 = patterns, for run-lengths A351292.
- A351202 = permutations of prime factors.
Selected statistics of standard compositions:
- Length is A000120.
- Parts are A066099, reverse A228351.
- Sum is A070939.
- Runs are counted by A124767, distinct A351014.
- Heinz number is A333219.
- Number of distinct parts is A334028.
Selected classes of standard compositions:
- Partitions are A114994, strict A333256.
- Multisets are A225620, strict A333255.
- Strict compositions are A233564.
- Constant compositions are A272919.
Cf. A098859, A106356, A113835, A116608, A238279, A242882, A318928, A325545, A328592, A329745, A350952, A351015, A351201.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],UnsameQ@@Split[stc[#]]&]

A351018 Number of integer compositions of n with all distinct even-indexed parts and all distinct odd-indexed parts.

Original entry on oeis.org

1, 1, 2, 3, 6, 9, 18, 27, 46, 77, 122, 191, 326, 497, 786, 1207, 1942, 2905, 4498, 6703, 10574, 15597, 23754, 35043, 52422, 78369, 115522, 169499, 248150, 360521, 532466, 768275, 1116126, 1606669, 2314426, 3301879, 4777078, 6772657, 9677138, 13688079, 19406214

Offset: 0

Gus Wiseman, Feb 09 2022

nonn

Also the number of binary words of length n starting with 1 and having all distinct runs (ranked by A175413, counted by A351016).

			The a(1) = 1 through a(6) = 18 compositions:
  (1)  (2)    (3)    (4)      (5)      (6)
       (1,1)  (1,2)  (1,3)    (1,4)    (1,5)
              (2,1)  (2,2)    (2,3)    (2,4)
                     (3,1)    (3,2)    (3,3)
                     (1,1,2)  (4,1)    (4,2)
                     (2,1,1)  (1,1,3)  (5,1)
                              (1,2,2)  (1,1,4)
                              (2,2,1)  (1,2,3)
                              (3,1,1)  (1,3,2)
                                       (2,1,3)
                                       (2,3,1)
                                       (3,1,2)
                                       (3,2,1)
                                       (4,1,1)
                                       (1,1,2,2)
                                       (1,2,2,1)
                                       (2,1,1,2)
                                       (2,2,1,1)

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

The case of partitions is A000726.
The version for run-lengths instead of runs is A032020.
These words are ranked by A175413.
A005811 counts runs in binary expansion.
A011782 counts integer compositions.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A059966 counts Lyndon compositions, necklaces A008965, aperiodic A000740.
A116608 counts compositions by number of distinct parts.
A238130 and A238279 count compositions by number of runs.
A242882 counts compositions with distinct multiplicities.
A297770 counts distinct runs in binary expansion.
A325545 counts compositions with distinct differences.
A329738 counts compositions with equal run-lengths.
A329744 counts compositions by runs-resistance.
A351014 counts distinct runs in standard compositions.
Counting words with all distinct runs:
- A351013 = compositions, for run-lengths A329739, ranked by A351290.
- A351016 = binary words, for run-lengths A351017.
- A351200 = patterns, for run-lengths A351292.
- A351202 = permutations of prime factors.
Cf. A003242, A025047, A098504, A098859, A106356, A212322, A328592, A329740, A334028, A349054, A350952, A351205.

Table[Length[Select[Tuples[{0,1},n],#=={}||First[#]==1&&UnsameQ@@Split[#]&]],{n,0,10}]

P(n)=prod(k=1, n, 1 + y*x^k + O(x*x^n));
seq(n)=my(p=P(n)); Vec(sum(k=0, n, polcoef(p,k\2,y)*(k\2)!*polcoef(p,(k+1)\2,y)*((k+1)\2)!)) \\ Andrew Howroyd, Feb 11 2022

a(n>0) = A351016(n)/2.

G.f.: Sum_{k>=0} floor(k/2)! * ceiling(k/2)! * ([y^floor(k/2)] P(x,y)) * ([y^ceiling(k/2)] P(x,y)), where P(x,y) = Product_{k>=1} 1 + y*x^k. - Andrew Howroyd, Feb 11 2022

Terms a(21) and beyond from Andrew Howroyd, Feb 11 2022

A351291 Numbers k such that the k-th composition in standard order does not have all distinct runs.

Original entry on oeis.org

13, 22, 25, 45, 46, 49, 53, 54, 59, 76, 77, 82, 89, 91, 93, 94, 97, 101, 102, 105, 108, 109, 110, 115, 118, 141, 148, 150, 153, 156, 162, 165, 166, 173, 177, 178, 180, 181, 182, 183, 187, 189, 190, 193, 197, 198, 201, 204, 205, 209, 210, 213, 214, 216, 217

Offset: 1

Gus Wiseman, Feb 12 2022

nonn

The n-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of n, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The terms together with their binary expansions and corresponding compositions begin:
  13:     1101  (1,2,1)
  22:    10110  (2,1,2)
  25:    11001  (1,3,1)
  45:   101101  (2,1,2,1)
  46:   101110  (2,1,1,2)
  49:   110001  (1,4,1)
  53:   110101  (1,2,2,1)
  54:   110110  (1,2,1,2)
  59:   111011  (1,1,2,1,1)
  76:  1001100  (3,1,3)
  77:  1001101  (3,1,2,1)
  82:  1010010  (2,3,2)
  89:  1011001  (2,1,3,1)
  91:  1011011  (2,1,2,1,1)
  93:  1011101  (2,1,1,2,1)
  94:  1011110  (2,1,1,1,2)

Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

The version for Heinz numbers of partitions is A130092, complement A130091.
Normal multisets with a permutation of this type appear to be A283353.
Partitions w/o permutations of this type are A351204, complement A351203.
The version using binary expansions is A351205, complement A175413.
The complement is A351290, counted by A351013.
A005811 counts runs in binary expansion, distinct A297770.
A011782 counts integer compositions.
A044813 lists numbers whose binary expansion has all distinct run-lengths.
A085207 represents concatenation of standard compositions, reverse A085208.
A333489 ranks anti-runs, complement A348612, counted by A003242.
A345167 ranks alternating compositions, counted by A025047.
Counting words with all distinct runs:
- A351016 = binary words, for run-lengths A351017.
- A351018 = binary expansions, for run-lengths A032020.
- A351200 = patterns, for run-lengths A351292.
- A351202 = permutations of prime factors.
Selected statistics of standard compositions (A066099, reverse A228351):
- Length is A000120.
- Sum is A070939.
- Runs are counted by A124767, distinct A351014.
- Heinz number is A333219.
- Number of distinct parts is A334028.
Selected classes of standard compositions:
- Partitions are A114994, strict A333256.
- Multisets are A225620, strict A333255.
- Strict compositions are A233564.
- Constant compositions are A272919.
Cf. A098859, A106356, A113835, A116608, A238279, A242882, A318928, A325545, A328592, A329745, A350952, A351015, A351201.

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],!UnsameQ@@Split[stc[#]]&]

A060142 Ordered set S defined by these rules: 0 is in S and if x is in S then 2x+1 and 4x are in S.

Original entry on oeis.org

0, 1, 3, 4, 7, 9, 12, 15, 16, 19, 25, 28, 31, 33, 36, 39, 48, 51, 57, 60, 63, 64, 67, 73, 76, 79, 97, 100, 103, 112, 115, 121, 124, 127, 129, 132, 135, 144, 147, 153, 156, 159, 192, 195, 201, 204, 207, 225, 228, 231, 240, 243, 249, 252, 255, 256, 259, 265, 268, 271

Offset: 0

Clark Kimberling, Mar 05 2001

nonn

After expelling 0 and 1, the numbers 4x occupy same positions in S that 1 occupies in the infinite Fibonacci word (A003849).
a(A026351(n)) = A219608(n); a(A004957(n)) = 4 * a(n). - Reinhard Zumkeller, Nov 26 2012
Apart from the initial term, this lists the indices of the 1's in A086747. - N. J. A. Sloane, Dec 05 2019
From Gus Wiseman, Jun 10 2020: (Start)
Numbers k such that the k-th composition in standard order has all odd parts, or numbers k such that A124758(k) is odd. The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. For example, the sequence of all compositions into odd parts begins:
()              57: (1,1,3,1)          135: (5,1,1,1)
(1)             60: (1,1,1,3)          144: (3,5)
(1,1)           63: (1,1,1,1,1,1)      147: (3,3,1,1)
(3)             64: (7)                153: (3,1,3,1)
(1,1,1)         67: (5,1,1)            156: (3,1,1,3)
(3,1)           73: (3,3,1)            159: (3,1,1,1,1,1)
(1,3)           76: (3,1,3)            192: (1,7)
(1,1,1,1)       79: (3,1,1,1,1)        195: (1,5,1,1)
(5)             97: (1,5,1)            201: (1,3,3,1)
(3,1,1)        100: (1,3,3)            204: (1,3,1,3)
(1,3,1)        103: (1,3,1,1,1)        207: (1,3,1,1,1,1)
(1,1,3)        112: (1,1,5)            225: (1,1,5,1)
(1,1,1,1,1)    115: (1,1,3,1,1)        228: (1,1,3,3)
(5,1)          121: (1,1,1,3,1)        231: (1,1,3,1,1,1)
(3,3)          124: (1,1,1,1,3)        240: (1,1,1,5)
(3,1,1,1)      127: (1,1,1,1,1,1,1)    243: (1,1,1,3,1,1)
(1,5)          129: (7,1)              249: (1,1,1,1,3,1)
(1,3,1,1)      132: (5,3)              252: (1,1,1,1,1,3)
(End)
Numbers whose binary representation has the property that every run of consecutive 0's has even length. - Harry Richman, Jan 31 2024

			From _Harry Richman_, Jan 31 2024: (Start)
In the following, dots are used for zeros in the binary representation:
   n  binary(a(n))  a(n)
   0:    .......     0
   1:    ......1     1
   2:    .....11     3
   3:    ....1..     4
   4:    ....111     7
   5:    ...1..1     9
   6:    ...11..    12
   7:    ...1111    15
   8:    ..1....    16
   9:    ..1..11    19
  10:    ..11..1    25
  11:    ..111..    28
  12:    ..11111    31
  13:    .1....1    33
  14:    .1..1..    36
  15:    .1..111    39
  16:    .11....    48
  17:    .11..11    51
  18:    .111..1    57
  19:    .1111..    60
  20:    .111111    63
  21:    1......    64
  22:    1....11    67
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Clark Kimberling, A Self-Generating Set and the Golden Mean, J. Integer Sequences, 3 (2000), #00.2.8.
Clark Kimberling, Fusion, Fission, and Factors, Fib. Q., 52 (2014), 195-202.
Lukasz Merta, Composition inverses of the variations of the Baum-Sweet sequence, arXiv:1803.00292 [math.NT], 2018. See l(n) p. 5.
Gus Wiseman, Statistics, classes, and transformations of standard compositions

Cf. A219608 (odd terms), A060138, A060139, A060140, A060141, A086747.
Cf. A003714 (no consecutive 1's in binary expansion).
Odd partitions are counted by A000009.
Numbers with an odd number of 1's in binary expansion are A000069.
Numbers whose binary expansion has odd length are A053738.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Compositions without odd parts are A062880.
- Sum is A070939.
- Product is A124758.
- Strict compositions are A233564.
- Heinz number is A333219.
- Number of distinct parts is A334028.

import Data.Set (singleton, deleteFindMin, insert)
a060142 n = a060142_list !! n
a060142_list = 0 : f (singleton 1) where
   f s = x : f (insert (4 * x) $ insert (2 * x + 1) s') where
       (x, s') = deleteFindMin s
-- Reinhard Zumkeller, Nov 26 2012

Take[Nest[Union[Flatten[# /. {{i_Integer -> i}, {i_Integer -> 2 i + 1}, {i_Integer -> 4 i}}]] &, {1}, 5], 32]  (* Or *)
Select[Range[124], FreeQ[Length /@ Select[Split[IntegerDigits[#, 2]], First[#] == 0 &], ?OddQ] &] (* _Birkas Gyorgy, May 29 2012 *)

is(n)=if(n<3, n<2, if(n%2,is(n\2),n%4==0 && is(n/4))) \\ Charles R Greathouse IV, Oct 21 2013

Corrected by T. D. Noe, Nov 01 2006

Definition simplified by Charles R Greathouse IV, Oct 21 2013

A345169 Numbers k such that the k-th composition in standard order is a non-alternating anti-run.

Original entry on oeis.org

37, 52, 69, 101, 104, 105, 133, 137, 150, 165, 180, 197, 200, 208, 209, 210, 261, 265, 274, 278, 300, 301, 308, 325, 328, 357, 360, 361, 389, 393, 400, 401, 406, 416, 417, 418, 421, 422, 436, 517, 521, 529, 530, 534, 549, 550, 556, 557, 564, 581, 600, 601, 613

Offset: 1

Gus Wiseman, Jun 15 2021

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.
A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,2,2,2,1) has no alternating permutations, even though it does have the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2).
An anti-run (separation or Carlitz composition) is a sequence with no adjacent equal parts.

			The sequence of terms together with their binary indices begins:
     37: (3,2,1)      210: (1,2,3,2)      400: (1,3,5)
     52: (1,2,3)      261: (6,2,1)        401: (1,3,4,1)
     69: (4,2,1)      265: (5,3,1)        406: (1,3,2,1,2)
    101: (1,3,2,1)    274: (4,3,2)        416: (1,2,6)
    104: (1,2,4)      278: (4,2,1,2)      417: (1,2,5,1)
    105: (1,2,3,1)    300: (3,2,1,3)      418: (1,2,4,2)
    133: (5,2,1)      301: (3,2,1,2,1)    421: (1,2,3,2,1)
    137: (4,3,1)      308: (3,1,2,3)      422: (1,2,3,1,2)
    150: (3,2,1,2)    325: (2,4,2,1)      436: (1,2,1,2,3)
    165: (2,3,2,1)    328: (2,3,4)        517: (7,2,1)
    180: (2,1,2,3)    357: (2,1,3,2,1)    521: (6,3,1)
    197: (1,4,2,1)    360: (2,1,2,4)      529: (5,4,1)
    200: (1,3,4)      361: (2,1,2,3,1)    530: (5,3,2)
    208: (1,2,5)      389: (1,5,2,1)      534: (5,2,1,2)
    209: (1,2,4,1)    393: (1,4,3,1)      549: (4,3,2,1)

Wikipedia, Alternating permutation

A version counting partitions is A345166, ranked by A345173.
These compositions are counted by A345195.
A001250 counts alternating permutations, complement A348615.
A003242 counts anti-run compositions.
A005649 counts anti-run patterns.
A025047 counts alternating or wiggly compositions, also A025048, A025049.
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A345164 counts alternating permutations of prime indices.
A345165 counts partitions w/o an alternating permutation, ranked by A345171.
A345170 counts partitions w/ an alternating permutation, ranked by A345172.
A345192 counts non-alternating compositions.
A345194 counts alternating patterns (with twins: A344605).
Statistics of standard compositions:
- Length is A000120.
- Constant runs are A124767.
- Heinz number is A333219.
- Anti-runs are A333381.
- Runs-resistance is A333628.
- Number of distinct parts is A334028.
- Non-anti-runs are A348612.
Classes of standard compositions:
- Weakly decreasing compositions (partitions) are A114994.
- Weakly increasing compositions (multisets) are A225620.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Strictly increasing compositions (sets) are A333255.
- Strictly decreasing compositions (strict partitions) are A333256.
- Anti-runs are A333489.
- Alternating compositions are A345167.
- Non-Alternating compositions are A345168.
Cf. A001222, A008965, A238279, A344614, A344615, A344652, A344653, A344654, A345162, A345163, A345193, A348609, A348613.

stc[n_]:=Differences[Prepend[Join@@Position[ Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]== Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
sepQ[y_]:=!MatchQ[y,{_,x_,x_,_}];
Select[Range[0,1000],sepQ[stc[#]]&&!wigQ[stc[#]]&]

Intersection of A345168 (non-alternating) and A333489 (anti-run).

A335238 Numbers k such that the distinct parts of the k-th composition in standard order (A066099) are not pairwise coprime, where a singleton is not coprime unless it is (1).

Original entry on oeis.org

0, 2, 4, 8, 10, 16, 32, 34, 36, 40, 42, 64, 69, 70, 81, 88, 98, 104, 128, 130, 136, 138, 139, 141, 142, 160, 162, 163, 168, 170, 177, 184, 197, 198, 209, 216, 226, 232, 256, 260, 261, 262, 274, 276, 277, 278, 279, 282, 283, 285, 286, 288, 290, 292, 296, 321

Offset: 1

Gus Wiseman, May 28 2020

nonn

We use the Mathematica definition for CoprimeQ, so a singleton is not considered coprime unless it is (1).

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The sequence together with the corresponding compositions begins:
    0: ()          88: (2,1,4)      177: (2,1,4,1)
    2: (2)         98: (1,4,2)      184: (2,1,1,4)
    4: (3)        104: (1,2,4)      197: (1,4,2,1)
    8: (4)        128: (8)          198: (1,4,1,2)
   10: (2,2)      130: (6,2)        209: (1,2,4,1)
   16: (5)        136: (4,4)        216: (1,2,1,4)
   32: (6)        138: (4,2,2)      226: (1,1,4,2)
   34: (4,2)      139: (4,2,1,1)    232: (1,1,2,4)
   36: (3,3)      141: (4,1,2,1)    256: (9)
   40: (2,4)      142: (4,1,1,2)    260: (6,3)
   42: (2,2,2)    160: (2,6)        261: (6,2,1)
   64: (7)        162: (2,4,2)      262: (6,1,2)
   69: (4,2,1)    163: (2,4,1,1)    274: (4,3,2)
   70: (4,1,2)    168: (2,2,4)      276: (4,2,3)
   81: (2,4,1)    170: (2,2,2,2)    277: (4,2,2,1)

The complement is A333228.
Not ignoring repeated parts gives A335239.
Singleton or pairwise coprime partitions are counted by A051424.
Singleton or pairwise coprime sets are ranked by A087087.
Coprime partitions are counted by A327516.
Non-coprime partitions are counted by A335240.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Sum is A070939.
- Product is A124758.
- Reverse is A228351
- GCD is A326674.
- Heinz number is A333219.
- LCM is A333226.
- Coprime compositions are A333227.
- Compositions whose distinct parts are coprime are A333228.
- Number of distinct parts is A334028.
Cf. A007360, A048793, A101268, A291166, A302569, A326675, A335235, A335236, A335237.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],!CoprimeQ@@Union[stc[#]]&]

A334440 Irregular triangle T(n,k) read by rows: row n lists numbers of distinct parts of the n-th integer partition in Abramowitz-Stegun (sum/length/lex) order.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 2, 2, 1, 3, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 2, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 2, 1, 3, 2, 3, 2, 2, 3, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 1, 3, 2, 2, 3, 3, 2, 2, 3, 3, 3, 3, 2, 2, 3

Offset: 0

Gus Wiseman, May 05 2020

The total number of parts, counting duplicates, is A036043. The version for reversed partitions is A103921.

			Triangle begins:
  0
  1
  1 1
  1 2 1
  1 1 2 2 1
  1 2 2 2 2 2 1
  1 1 2 2 1 3 2 2 2 2 1
  1 2 2 2 2 2 3 2 2 3 2 2 2 2 1
  1 1 2 2 2 2 2 3 3 2 1 3 2 3 2 2 3 2 2 2 2 1

Robert Price, Table of n, a(n) for n = 0..9295 (25 rows).
Wikiversity, Lexicographic and colexicographic order

Row lengths are A000041.
The number of not necessarily distinct parts is A036043.
The version for reversed partitions is A103921.
Ignoring length (sum/lex) gives A103921 (also).
a(n) is the number of distinct elements in row n of A334301.
The maximum part of the same partition is A334441.
Lexicographically ordered reversed partitions are A026791.
Reversed partitions in Abramowitz-Stegun (sum/length/lex) order are A036036.
Partitions in increasing-length colex order (sum/length/colex) are A036037.
Graded reverse-lexicographically ordered partitions are A080577.
Partitions counted by sum and number of distinct parts are A116608.
Graded lexicographically ordered partitions are A193073.
Partitions in colexicographic order (sum/colex) are A211992.
Partitions in dual Abramowitz-Stegun (sum/length/revlex) order are A334439.
Cf. A001221, A049085, A124734, A185974, A296774, A299755, A334028, A334302, A334433, A334434, A334435, A334438.

Join@@Table[Length/@Union/@Sort[IntegerPartitions[n]],{n,0,10}]

a(n) = A001221(A334433(n)).

cp's OEIS Frontend

A351015 Smallest k such that the k-th composition in standard order has n distinct runs.

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A351016 Number of binary words of length n with all distinct runs.

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A351596 Numbers k such that the k-th composition in standard order has all distinct run-lengths.

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A351290 Numbers k such that the k-th composition in standard order has all distinct runs.

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A351018 Number of integer compositions of n with all distinct even-indexed parts and all distinct odd-indexed parts.

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A351291 Numbers k such that the k-th composition in standard order does not have all distinct runs.

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A060142 Ordered set S defined by these rules: 0 is in S and if x is in S then 2x+1 and 4x are in S.

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A345169 Numbers k such that the k-th composition in standard order is a non-alternating anti-run.